Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488707 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #466 | C23 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 667 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(4,11)(6,12)(7,14)(9,15)(17,18,25,32)(19,33,39,65)(20,40,44,63)(21,29,46,59)(22,28,45,60)(23,31,24,30)(26,50,55,36)(27,49,58,43)(34,66,70,77)(35,54,47,53)(37,61,38,52)(41,56,75,74)(42,62,48,57)(51,67,80,76)(64,72,78,69)(68,73,71,79);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,25,39)(18,26,32,55)(20,34,44,70)(21,36,46,50)(22,47,45,35)(23,38,24,37)(27,51,58,80)(28,33,60,65)(29,61,59,52)(30,54,31,53)(40,73,63,79)(41,71,75,68)(42,67,48,76)(43,72,49,69)(56,64,74,78)(57,77,62,66);(1,3,5,10)(2,6,8,12)(4,9,11,15)(7,13,14,16)(17,20,25,44)(18,27,32,58)(19,34,39,70)(21,41,46,75)(22,48,45,42)(23,49,24,43)(26,51,55,80)(28,56,60,74)(29,62,59,57)(30,63,31,40)(33,64,65,78)(35,67,47,76)(36,71,50,68)(37,72,38,69)(52,77,61,66)(53,73,54,79);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,24,45,25,46,23,22)(18,28,31,59,32,60,30,29)(19,35,38,36,39,47,37,50)(20,41,43,42,44,75,49,48)(26,52,54,33,55,61,53,65)(27,56,40,57,58,74,63,62)(34,67,69,71,70,76,72,68)(51,77,79,64,80,66,73,78)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396015);